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Kurt Gödel

Kurt Gödel was born on April 28, 1906 in what was then the Austro-Hungarian city of Brünn, and what is now Brno in the Czech Republic.

Gödel’s father Rudolf August was a businessman, and his mother Marianne was a well-educated and cultured woman to whom Gödel remained close throughout his life, as witnessed by the long and wide-ranging correspondence between them. The family was well off, and Gödel’s childhood was an uneventful one, with one important exception; namely, from about the age of four Gödel suffered frequent episodes of poor health, and the health problems he suffered then as well as others of various kinds were to plague him his entire life.

Health problems notwithstanding, Gödel proved to be an exemplary student at primary school and later the Gymnasium, excelling especially in mathematics, languages and religion. Upon his graduation from the Gymnasium in Brno in 1924 Gödel enrolled in the University of Vienna, attending lectures on physics, his initial field of interest, lectures on philosophy given by Heinrich Gomperz, and lectures on mathematics. Gödel took a substantial numbers of physics courses during his undergraduate years, as witnessed by his university transcript; this is notable in view of Gödel’s subsequent contributions to relativity in 1947. Philipp Furtwängler, cousin of the great German conductor Wilhelm Furtwängler, was one of his mathematics professors, and indeed Furtwängler’s course on class field theory almost tempted Gödel to pursue his studies in that area. Gödel learned his logic from Rudolph Carnap and from Hans Hahn, eventually graduating under Hahn with a Dr.phil. in mathematics in 1929. The main theorem of his dissertation was the completeness theorem for first order logic (Gödel 1929).[2]

Gödel’s university years also marked the beginning of his attendance at meetings of the Vienna Circle, a group around Moritz Schlick that quickly became known as “logical positivists,” a term coined by Feigl and Blumberg in their 1931 “Logical positivism: A new movement in European philosophy” (Feigl and Blumberg 1931). Though Gödel was not himself a logical positivist, those discussions must have been a crucial formative influence.

The 1930s were a prodigious decade for Gödel, while at the same time being also quite a turbulent period for Gödel personally. After publishing the paper based on his 1929 dissertation in 1930, he published his groundbreaking incompleteness theorems in 1931, on the basis of which he was granted his Habilitation in 1932 and a Privatdozentur at the University of Vienna in 1933.

Among his mathematical achievements at the decade’s close is the proof of the consistency of both the Axiom of Choice and Cantor’s Continuum Hypothesis with the Zermelo-Fraenkel axioms for set theory, obtained in 1935 and 1937, respectively. Gödel also published a number of significant papers on modal and intuitionistic logic and arithmetic during this period. Principal among the latter is Gödel’s “On intuitionistic arithmetic and number theory,” (Gödel 1933e), in which Gödel showed that classical first order arithmetic is interpretable in Heyting arithmetic by a simple translation. The 1930s also saw the publication by Gödel on a wide range of other topics in logic and mathematics, ranging from the decision problem for the predicate calculus, to those on the length of proofs, to those on differential and projective geometry.

On the other hand, by the end of the decade both Gödel’s advisor Hans Hahn and Moritz Schlick had died (the latter was assassinated by an ex-student), two events which led to a personal crisis for Gödel (as would the death of Einstein in 1955). Also, difficulties emerged concerning his appointment at the University of Vienna: the Nazis abolished the old position of Privatdozentur, replacing it by the position “Dozentur neuer Ordnung,” granted to candidates only after they had passed a political test.[3] Gödel’s three trips the United States during that period triggered a similar investigation. (See Sigmund 2006.) Finally, Gödel was found fit for military service by the Nazi government in 1939.

All of these events were undoubtedly decisive in influencing his decision to leave Austria, which he and his wife Adele were able to do in 1940, when they immigrated to the United States. This long and difficult episode in their life is recounted by John Dawson in his biography of Gödel called “Logical Dilemmas,” (Dawson 1997) as well as by Solomon Feferman in “Gödel’s Life and Work,” (Feferman 1986) to both of which the reader is referred.

Upon arrival Gödel took up an appointment as an ordinary member at the Institute for Advanced Study; he would become a permanent member of the Institute in 1946 and would be granted his professorship in 1953. (Gödel and his wife were granted American citizenship in April 1948.) He would remain at the Institute until his retirement in 1976. The Gödels never returned to Europe.

Gödel’s early years at the Institute were notable for his close friendship with his daily walking partner Albert Einstein, as well as for his turn to philosophy of mathematics, a field on which Gödel began to concentrate almost exclusively from about 1943. The initial period of his subsequent lifelong involvement with philosophy was a fruitful one (in terms of publications): in 1944 he published his first philosophical paper, entitled “On Russell’s Mathematical Logic” (Gödel 1944), and in 1947 he published his second, entitled “What is Cantor’s Continuum Hypothesis?” (Gödel 1947). In 1949 he published his third, entitled “A Remark on the Relationship between Relativity Theory and Idealistic Philosophy.” (Gödel 1949a). The latter paper coincided with results on rotating universes in relativity he had obtained in 1949, which were first published in an article entitled: “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation.” (Gödel 1949).

Among Gödel’s other significant philosophical works of the 1940’s must be counted his 1941 lecture at Yale entitled “In What Sense is Intuitionistic Logic Constructive?” (Gödel *1941) in which the notion: “computable function of finite type” is introduced. A paper based on the ideas in the lecture entitled “Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes,” was published only in 1958, and the interpretation of Heyting arithmetic into the quantifier free calculus T in it became known as the “Dialectica Interpretation,” after the journal in which the article was published (Gödel 1958). (For the revision of it from 1972, see Gödel 1995.) Finally the decade saw the beginning of Gödel’s intensive study of Leibniz, which, Gödel reports, occupied the period from 1943 to 1946.[4]

The 1950s saw a deepening of Gödel’s involvement with philosophy: In 1951 Gödel delivered a philosophical lecture at Brown University, usually referred to as the Gibbs Lecture, entitled “Some Basic Theorems on the Foundations of Mathematics and Their Philosophical Implications” (Gödel *1951). From 1953 to 1959 Gödel worked on a submission to the Schilpp volume on Rudolf Carnap entitled “Is Mathematics a Syntax of Language?” (Gödel *1953/9-III, Gödel *1953/9-V). Gödel published neither of these two important manuscripts in his lifetime, although both would appear on two lists which were found in the Gödel Nachlass, entitled “Was ich publizieren könnte.” (In English: “What I could publish.” Both manuscripts eventually appeared in Gödel 1995.) By the decade’s close Gödel underwent a significant change in his thinking, converting to Husserlian phenomenology as a means to systematize the Leibnizian views that had been in place since at least the 1930s.[5]

Gödel’s final years are notable for his circulation of two manuscripts: “Some considerations leading to the probable conclusion that the true power of the continuum is ℵ2,” (Gödel *1970a, *1970b) his attempt to derive the value of the continuum from the so-called scale axioms of Hausdorff, and his “ontologischer Beweis,” (Gödel *1970) which he entrusted to Dana Scott in 1970 (though it appears to have been written earlier). Taken together, the two manuscripts are the fitting “last words” of someone who, in a fifty year involvement with mathematics and philosophy, pursued, or more precisely, sought the grounds for pursuing those two subjects under the single heading: “strenge Wissenschaft”—an attitude, or turn of mind, or wish, if you will, Gödel held from his start in 1929, when at the age of twenty-three he opened his doctoral thesis with some philosophical remarks.

Gödel died in Princeton on January 14, 1978 at the age of 71. His death certificate records the cause of death, tragically, as “starvation and inanition, due to personality disorder.” His wife Adele survived him by three years.

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1 Comment

Yes, Godel had some unusual, and interesting things to say about the “other world,” but his excellent ideas (and those of others) are incomplete if they do not take into account the development of Multi-Dimensional Science, or MDS for short. This is my project which if “correct” could ultimately have profound, and revolutionary implications for the world.